QR code can be spoofed and that happened already - there were certificates issued for e.g. Adolf Hitler and also for Mickey Mouse with govt health agency signature [1]; these were using French and German keys. Both are invalidated of course. And not mention all "classic" written physical information cards - these are around too for people who want to avoid vaccines for whatever reasons they want.
On a paper this looks great - hell, the solution might even work for a time but additional unexpected factors may spawn out of nowhere and destroy whole thing.
The qr code idea in Europe works together with your id. Americans don't want to use ID even for voting, hard to expect to have working vaccination card program. Hence they're left with random papers and the problem on hand.
Pretty sure the US doesn't have an official federal-issued ID, and most of the time driving license works as such. And it seems this applies to the majority of Anglosphere countries.
Here in Poland we have a compulsory IDs, which for over 20 years now are plastic. Recently RFID tags and finger prints were added; there's also a standard biometric photo of face. All of this comes also with electronic and qualified electronic signature which can be used to verify our identity in government services or to sign the digital documents. And yes, we need to have it or the slowly accepted digital version when we want to vote in any elections.
Having a federal ID might make sense if you're voting in federal elections, but there are no direct federal elections in the US. People in the US vote in state elections, and the states then vote in the electoral college.
You wouldn't need a federal-issued ID. You can use the state id which you also use to identify yourself for everything else. National ID "requirement" is a fake obstacle. Identification verification is already a requirement as well - you must give your name and attest that you are who you say you are. Lying about it is a crime. An ID would not add an additional requirement, it would just enable better enforcement.
Besides, with "Real ID" state ID's meet federal verification standards now as well.
That's funny, considering the concept of a national ID has been soundly rejected by republicans for decades. Instead we shoehorned SSN's into that position.
Most Republicans do not favor national ID's and do favor checking ID's to vote. The idea behind voter ID is using the same state ID you use to buy liquor, etc.